In this advanced probability course, we will explore several stochastic models arising in the mathematical theory of population genetics.

The main goal of theoretical population genetics, pioneered by the works of Fisher, Haldane, and Wright in the early 20th century, is to understand the genetic variation that can be observed in the real world. The theory provided the framework that successfully reconciled Darwinian evolution with Mendelian genetics.

In this course we will study several stochastic population models under the action of different evolutionary forces. Our guiding questions, which we will tackle using the theory of Markov processes, are:
(Forward in time) How does the stochastic allele-frequency process evolve?
(Backward in time) What is the genealogy of a sample from the current population?

Our starting point is the fundamental Wright–Fisher model, introduced to describe the phenomenon of random genetic drift. We will then consider further evolutionary forces, such as selection and mutation, as well as more complicated structured models. Along the way, we will study scaling limits and duality, with particular emphasis on the connection between forward-in-time population dynamics and backward-in-time genealogical processes. As we will see, these tools are relevant to a broad class of interacting particle systems.

Keywords: Wright–Fisher model, Moran model, Kingman coalescent, Lambda coalescent, Xi coalescent, duality, Markov processes, interacting particle systems.